1. **State the problem:** Simplify the expression $$4 \sqrt[3]{7} + 9 \sqrt[3]{56}$$. 2. **Recall the formula and rules:** To simplify sums involving cube roots, express all terms with the same radicand if possible, then combine coefficients. 3. **Simplify the radicand in the second term:** Note that $$56 = 7 \times 8$$, and $$8 = 2^3$$ is a perfect cube. 4. Rewrite $$\sqrt[3]{56}$$ as $$\sqrt[3]{7 \times 8} = \sqrt[3]{7} \times \sqrt[3]{8} = \sqrt[3]{7} \times 2$$. 5. Substitute back into the expression: $$4 \sqrt[3]{7} + 9 \times 2 \sqrt[3]{7} = 4 \sqrt[3]{7} + 18 \sqrt[3]{7}$$. 6. Combine like terms: $$4 \sqrt[3]{7} + 18 \sqrt[3]{7} = (4 + 18) \sqrt[3]{7} = 22 \sqrt[3]{7}$$. **Final answer:** $$22 \sqrt[3]{7}$$.